Application of the perfectly matched layer (PML) absorbing boundary condition to elastic wave propagation
01 Nov 1996-Journal of the Acoustical Society of America (Acoustical Society of America)-Vol. 100, Iss: 5, pp 3061-3069
TL;DR: Peng and Toksoz as mentioned in this paper presented a method for application of the perfectly matched layer absorbing boundary condition (ABC) to the P•SV velocity-stress finite-difference method.
Abstract: A method is presented for application of the perfectly matched layer (PML) absorbing boundary condition (ABC) to the P‐SV velocity–stress finite‐difference method The PML consists of a nonphysical material, containing both passive loss and dependent sources, that provides ‘‘active’’ absorption of fields It has been used in electromagnetic applications where it has provided excellent results for a wide range of angles and frequencies In this work, numerical simulations are used to compare the PML and an ‘‘optimal’’ second‐order elastic ABC [Peng and Toksoz, J Acoust Soc Am 95, 733–745 (1994)] Reflection factors are used to compare angular performance for continuous wave illumination; snapshots of potentials are used to compare performance for broadband illumination These comparisons clearly demonstrate the superiority of the PML formulation Within the PML there is a 60% increase in the number of unknowns per grid cell relative to the velocity–stress formulation However, the high quality of the PML ABC allows the use of a smaller grid, which can result in a lower overall computational cost
Citations
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TL;DR: In this article, a perfectly matched absorbing layer model for the velocity-stress formulation of elastodynamics is proposed, which decomposes each component of the unknown into two auxiliary components: a component orthogonal to the boundary and a component parallel to it.
Abstract: We present and analyze a perfectly matched, absorbing layer model for the velocity-stress formulation of elastodynamics. The principal idea of this method consists of introducing an absorbing layer in which we decompose each component of the unknown into two auxiliary components: a component orthogonal to the boundary and a component parallel to it. A system of equations governing these new unknowns then is constructed. A damping term finally is introduced for the component orthogonal to the boundary. This layer model has the property of generating no reflection at the interface between the free medium and the artificial absorbing medium. In practice, both the boundary condition introduced at the outer boundary of the layer and the dispersion resulting from the numerical scheme produce a small reflection which can be controlled even with very thin layers. As we will show with several experiments, this model gives very satisfactory results; namely, the reflection coefficient, even in the case of heterogeneous, anisotropic media, is about 1% for a layer thickness of five space discretization steps.
739 citations
Cites methods from "Application of the perfectly matche..."
...In Hastings et al. (1996), the authors propose the use of PML for the compressional and shear potentials formulation....
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TL;DR: In this article, the authors demonstrate how to improve the perfectly matched layer (PML) absorbing boundary condition at grazing incidence for the differential seismic wave equation based on an unsplit convolution technique.
Abstract: The perfectly matched layer (PML) absorbing boundary condition has proven to be very efficient from a numerical point of view for the elastic wave equation to absorb both body waves with nongrazing incidence and surface waves. However, at grazing incidence the classical discrete PML method suffers from large spurious reflections that make it less efficient for instance in the case of very thin mesh slices, in the case of sources located close to the edge of the mesh, and/or in the case of receivers located at very large offset. We demonstrate how to improve the PML at grazing incidence for the differential seismic wave equation based on an unsplit convolution technique. The improved PML has a cost that is similar in terms of memory storage to that of the classical PML. We illustrate the efficiency of this improved convolutional PML based on numerical benchmarks using a finite-difference method on a thin mesh slice for an isotropic material and show that results are significantly improved compared with the classical PML technique. We also show that, as the classical PML, the convolutional technique is intrinsically unstable in the case of some anisotropic materials.
659 citations
Cites methods from "Application of the perfectly matche..."
...Regarding seismic wave propagation, the PML has been successfully applied to both acoustice.g., Liu and Tao, 1997; Qi and Geers, 1998; Abarbanel et al., 1999; Katsibas and Antonopoulos, 2002; Diaz and Joly, 2006; Bermudez et al., 2007 and elastic problems e.g., Chew and Liu, 1996; Hastings et al., 1996; Collino and Tsogka, 2001; Festa and Nielsen, 2003; Komatitsch and Tromp, 2003; Basu and Chopra, 2004; Rahmouni, 2004; Cohen and ......
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TL;DR: In this article, a necessary and sufficient condition for the stability of the perfectly-matched layers (PML) model for a general hyperbolic system is derived from the geometrical properties of the slowness diagrams.
383 citations
Cites background from "Application of the perfectly matche..."
...), including in particular elastic wave propagation in isotropic [21] and anisotropic media [17]....
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TL;DR: The perfectly matched layer absorbing boundary condition has proven to be very efficient for the elastic wave equation written as a first-order system in velocity and stress as mentioned in this paper, which facilitates use in the context of numerical schemes based upon such a system, e.g. the finite element method, the spectral element method and some finite difference methods.
Abstract: SUMMARY
The perfectly matched layer absorbing boundary condition has proven to be very efficient for the elastic wave equation written as a first-order system in velocity and stress. We demonstrate how to use this condition for the same equation written as a second-order system in displacement. This facilitates use in the context of numerical schemes based upon such a system, e.g. the finite-element method, the spectral-element method and some finite-difference methods. We illustrate the efficiency of this second-order perfectly matched layer based upon 2-D benchmarks with body and surface waves.
371 citations
Cites methods from "Application of the perfectly matche..."
...In the context of wave propagation, the PML has been applied to both acoustic (e.g. Liu & Tao 1997; Qi & Geers 1998; Hagstrom 1999) and elastic problems (e.g. Chew & Liu 1996; Hastings et al. 1996; Collino & Monk 1998a; Collino & Tsogka 2001; Basu & Chopra 2003; Cohen & Fauqueux 2003)....
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TL;DR: In this article, a displacement-based, symmetric finite-element implementation of the perfectly matched layer (PML) model is presented for time-harmonic plane-strain or three-dimensional motion.
334 citations
Cites methods from "Application of the perfectly matche..."
...[29] applied B erenger s original split-field formulation of the electromagnetics PML directly to the P- and S-wave potentials and obtained a two-dimensional FDTD scheme for implementing the resultant formulation....
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References
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TL;DR: Numerical experiments and numerical comparisons show that the PML technique works better than the others in all cases; using it allows to obtain a higher accuracy in some problems and a release of computational requirements in some others.
9,875 citations
TL;DR: In this paper, a finite-difference method for modeling P-SV wave propagation in heterogeneous media is presented, which is an extension of the method I previously proposed for modeling SH-wave propagation by using velocity and stress in a discrete grid, where the stability condition and the P-wave phase velocity dispersion curve do not depend on the Poisson's ratio.
Abstract: I present a finite-difference method for modeling P-SV wave propagation in heterogeneous media This is an extension of the method I previously proposed for modeling SH-wave propagation by using velocity and stress in a discrete grid The two components of the velocity cannot be defined at the same node for a complete staggered grid: the stability condition and the P-wave phase velocity dispersion curve do not depend on the Poisson's ratio, while the S-wave phase velocity dispersion curve behavior is rather insensitive to the Poisson's ratio Therefore, the same code used for elastic media can be used for liquid media, where S-wave velocity goes to zero, and no special treatment is needed for a liquid-solid interface Typical physical phenomena arising with P-SV modeling, such as surface waves, are in agreement with analytical results The weathered-layer and corner-edge models show in seismograms the same converted phases obtained by previous authors This method gives stable results for step discontinuities, as shown for a liquid layer above an elastic half-space The head wave preserves the correct amplitude Finally, the corner-edge model illustrates a more complex geometry for the liquid-solid interface As the Poisson's ratio v increases from 025 to 05, the shear converted phases are removed from seismograms and from the time section of the wave field
2,583 citations
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04 Sep 2011TL;DR: In this paper, a discussion of the behavior of the solution as the mesh width tends to zero is presented, and the applicability of the method to more general difference equations and to those with arbitrarily many independent variables is made clear.
Abstract: Problems involving the classical linear partial differential equations of mathematical physics can be reduced to algebraic ones of a very much simpler structure by replacing the differentials by difference quotients on some (say rectilinear) mesh. This paper will undertake an elementary discussion of these algebraic problems, in particular of the behavior of the solution as the mesh width tends to zero. For present purposes we limit ourselves mainly to simple but typical cases, and treat them in such a way that the applicability of the method to more general difference equations and to those with arbitrarily many independent variables is made clear.
2,047 citations
TL;DR: A modified set of Maxwell's equations is presented that includes complex coordinate stretching along the three Cartesian coordinates that allow the specification of absorbing boundaries with zero reflection at all angles of incidence and all frequencies.
Abstract: A modified set of Maxwell's equations is presented that includes complex coordinate stretching along the three Cartesian coordinates. The added degrees of freedom in the modified Maxwell's equations allow the specification of absorbing boundaries with zero reflection at all angles of incidence and all frequencies. The modified equations are also related to the perfectly matched layer that was presented recently for 2D wave propagation. Absorbing-material boundary conditions are of particular interest for finite-difference time-domain (FDTD) computations on a single-instruction multiple-data (SIMD) massively parallel supercomputer. A 3D FDTD algorithm has been developed on a connection machine CM-5 based on the modified Maxwell's equations and simulation results are presented to validate the approach. © 1994 John Wiley & Sons, Inc.
1,660 citations
TL;DR: The Madariaga-Virieux staggered-grid scheme has the desirable quality that it can correctly model any variation in material properties, including both large and small Poisson's ratio materials, with minimal numerical dispersion and numerical anisotropy.
Abstract: I describe the properties of a fourth-order accurate space, second-order accurate time two-dimensional P-Sk’ finite-difference scheme based on the MadariagaVirieux staggered-grid formulation. The numerical scheme is developed from the first-order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses. The Madariaga-Virieux staggered-grid scheme has the desirable quality that it can correctly model any variation in material properties, including both large and small Poisson’s ratio materials, with minimal numerical dispersion and numerical anisotropy. Dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints/wavelength. The scheme can be used to accurately simulate wave propagation in mixed acoustic-elastic media, making it ideal for modeling marine problems. Explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free-surface or within a layer and to satisfy free-surface boundary conditions. Benchmark comparisons of finite-difference and analytical solutions to Lamb’s problem are almost identical, as are comparisons of finite-difference and reflectivity solutions for elastic-elastic and acoustic-elastic layered models.
1,429 citations